'The Big Blue' exists in three different versions: the European theatrical version, the American theatrical version, and the European extended version, which first was released on VHS and later on DVD, too. However, there are no censoring cuts when you compare the European theatrical version with the extended version.
As biggest difference between the two versions, the Version Longue contains a couple more Story elements which slow down the movie a bit, but gains additional depth concerning characters. Additionally, some new characters which were not in the theatrical release appear.
Speaking of theaters: In the American theatrical release, the movie has an alternative ending. Jacques is being rescued by the dolphin instead of committing his suicide in the dark depths of the big blue. This ending can be watched on YouTube (search 'The Big Blue USA version ending'). In this clip, you can also hear the soundtrack from Bill Conti which had been created for the American version.

But this version is not subject to this report, which will focus on the comparison between the European theatrical version ('TV', represented by the English VHS tape) and the Version Longue ('VL', represented by the German extended version on DVD).

The VHS version has an 4:3 aspect ratio. During the opening credits, you can see the movie in its original scope format, but after 2.44 mins., the picture is scaled up to 4:3. The framing is lost, together with parts of the picture. In some scenes, you have a movement of the picture via Pan & Scan, which are no camera movements the director had intended. Two examples for the different formats:

00.00New animated Gaumont sign in place of the old 20th Century Fox sign.
+11 secs.

No runtime differenceDuring the beginning, the opening credits are different. The VL contains more names.
0 secs.

03.48A long shot of the Greek village on the island has been inserted.
+5 secs.

04.07The kids are running a bit longer.
+1 sec.

04.08Five additional shots of the children running through the village. Some of the credits' text dis- and reappears behind the running children.
+56 secs.

05.04The cut begins a bit earlier.
+1 sec.

05.17Jaques ends the brawl of the kids. You get a short shot of the priest. Jacques proposes to get the coin and share it afterwards. Someone asks since when coins were shareable? Proposition of Jacques: Buy something beautiful and share that.
+21 secs.

07.00After Enzo instead of Jacques has brought the coin out of the water, a discussion about the ownership of the coin starts. Enzo now for the first time suggests the challenge of a diving contest. Jacques refuses and Enzo and his gang disappear without sharing the coin.
+46 secs.

08.50Jacques is feeding the moray a bit longer.
+8 secs.

12.05A short dialog between Jacques and the fisherman asking if Jaques ever had seen a mermaid. This dialog doesn't exist in the TV.
+5 secs.

12.20Now the viewer learns that Jacques' mother had left his father. Jacques asks why. In the TV, there are three short shots of Enzo, Jacques, and his father underwater.
+40 secs.

15.59Enzo, Roberto, and the American can be seen from behind, followed by a shot where all three of them are looking towards the shipwreck.
+4 secs.

16.29The American feels hosed and loses his temper. Enzo and Roberto walk towards him and ask what a man's life is worth to him.
+31 secs.

20.31The American is following Enzo and his brother and thanks them for saving him.
+28 secs.

38.20A shot of Johana and her friend is added.
+14 secs.

38.46New York at night.
+4 secs.

38.50Johana is lying on her bed and tells her (visibly bored) friend about Jacques.
+52 secs.

00.00This scene is missing in the VL: Searching for Jacques, Johana exits the elevator in the hotel and walks to the restaurant through the corridor in the rock.
-13 secs.

44.43The dialog between Enzo and Jacques lasts a bit longer. They talk about the past years in which they hadn't seen each other.
+51 secs.

55.26Enzo and Jacques are talking with each other a bit longer. Enzo claims to be unbeatable in diving and that Jacques might be too frail for diving.
+35 secs.

1.00.52The doctor examines Enzo and severely advises him not to dive that day. Enzo doesn't accept this decision and clearly staes this towards the doctor. He says that only he himself may decide when the sea will take him. Enzo doesn't listen to the doctor's warnings.
+42 secs.

1.06.17At this point, a whole new sequence has been inserted: the committee wants to honor Enzo's new world record with a trophy, but he is in his room and doesn't show up. Jacques appears and is asked to help. When Jacques knocks on Enzo's door and tells his name, he is received heartily by Enzo. The clearly annoyed committee asks
Enzo once more to attend the ceremony. Enzo says they should tell the press that he is not well. He shuts the door close before their astonished faces. Jacques asks him why he doesn't want a trophy. Enzo wants to eat pasta instead.
+65 secs.

1.20.13Jacques and Johana get off the boat (this shot was added by fading).
+10 secs.

1.23.58Jacques is watching the dolphin at sea a bit longer. There's an additional shot of the dolphin, too. Contrary to the TV, the shot of Johana lying in bed and calling for Jacques (audible in the background as well) is missing. That's why Jacques' reason for turning around has changed here: in the TV, he walks back to Johana. In the VL, Jacques walks to the beach and the dolphin as Johana hasn't recognized him being on the balcony and watching the sea.
+6 secs.

In the TV, the following chapters are missing. The TV jumps directly from the balcony at night (Jacques returning to Johana in the bedroom) to the French Riviera (the following cut takes us directly to Jacques' apartment). This leap in time is covered by two 'establishing shots' in the TV.
-9 secs.

01.24.10Here, several chapters are inserted: Jacques is swimming with the dolphins in the VL. Johana doesn't recognize his leaving until morning. She is worriedly waiting for him on the beach. She also lacks sympathy for his actions. The same day, they say goodbye to each other at the train station.

Jacques asks Enzo for advice. Enzo suggests Jacques work with him as diver on an oil rig. Jacques accepts. Enzo and Jacques however don't take their jobs too seriously. In the pressure chamber, they philosophy about women and drink alcohol. Then they waltz through the water, drunk. This leads to the loss of their jobs.

Johana - back in New York - is heartsick, but her friend doesn't listen to her. Further, she has to tell her boss that the job in Sicily had been a lie and that she had gone there to see Jacques. Johana, too, can kiss her job goodbye. She talks with Jacques a long time over phone and they agree to meet again.

Jacques picks up Johana at the airport. On the way to the hotel, they already start making out in the taxi. In the hotel room, they sleep together very passionately. Jacques, however, repeatedly gets flashes of the deep sea. The next day, Jaques takes Johana to his apartment.

+18 mins. 42 secs.

1.43.11Jacques takes Johana to his apartment, where they encounter uncle Loui. Loui is diving in his bathtub while listening to Wagner's 'Ride of the Valkyries' underwater. During the meal, Jacues introduces Johana once more and tells his cranky uncle that she will stay with them for a bit. Uncle Loui reluctantly notices this fact.
+155 secs.

01.46.50Here, the script was changed, so the VL contains a slightly shorter shot of Johana.
-1.5 secs.

1.47.14Jacques, Johanna, and Roberto are waiting for Enzo. Uncle Loui complains about the high fish price. Enzo arrives via speedboat and introduces his new girlfriend Bonita, a Spanish actress. Afterwards, they party in Jacques' apartment. Doctor Lawrence drops by, too, and meets Enzo for the first time. Meanwhile, Johana and Bonita are talking about their boyfriends and possible children.
+204 secs.

1.51.22Enzo has problems at his dive. Him being distracted by his girlfriend, the controllers at one waypoint refuse that he may dive deeper. Angrily, he ascends again.
+71 secs.

1.46.04Once again, complete sequences have been inserted. You can see the other divers in competition, trying to be like Enzo. But hey all fail. The Japanese team doesn't even get their turn as their diver starts hyperventilating. Enzo and Roberto are watching the scene with malicious joy. Jacques asks about Enzo's record and is warned by the jury that it's getting dangerous now. He is impressed and congratulates Enzo. At the award ceremony, everyone congratulates and the jury once more warns that depths are getting dangerous now. However, the competition will be continued.
+14 Min 28 secs.

2.00.52The end of a scene is a bit later, as there are some shots of Jaques' preparations for the dive before he starts the actual dive. The shot when Jacques gets counted in for diving starts a bit earlier.
+79 secs.

2.05.40Bonita says goodbye to Johana. She leaves Enzo and wishes good luck to Johana.
+71 secs.

2.08.52Johana is sitting a bit longer. They both move into their room. Johana visibly tired. On the porch, she asks Jacques about his feelings when he's underwater. Jacques says that it is hard for him to find a reason every time to get to the surface again. Johana feels similarly. She is searching for a reason to stay with Jacques. Jacques kisses Johana and she says that she has found the reason now.
+129 secs.

2.14.00Johana cries for Jacques who doesn't reappear from the water.
+9 secs.

2.22.55Jacques gets CPR after he has brought his dead friend Enzo back under water. Meanwhile, Jacques is hallucinating of his dolphins. The doctors manage to keep him alive.
+83 secs.

2.25.43Roberto blames himself that he hasn't been on the boat when Enzo died.
+49 secs.

2.32.58Johana tells Jacques that she is pregnant.
+7 secs.

2.33.54An overhead shot is inserted. Johana is holding his arm a bit longer (in the TV, Jacques now tells Johana that he loves her. In the VL, this declaration of love is missing.)
+7 secs.